Example 1.11 
Let \(\mathfrak {g}\)  be the 6-dimensional solvable Lie algebra in [1]}. We do not need the whole definition and use only the fact that both \(\mathfrak {g}/\mathfrak {e}\)  and \(\mathfrak {e}\)  are isomorphic to the 3-dimensional Heisenberg Lie algebra.
Denote by \(G\)  the associated simply connected Lie group. Consider the coordinates \((s_1, s_2, s_3, t_1, t_2, t_3)\)  on \(G\) , corresponding to the canonical coordinates of the first kind on both copies of the Heisenberg algebra (first three for \(\mathfrak {g}/\mathfrak {e}\)  and last three for \(\mathfrak {e}\) ; see details in the above reference). Then each word length function on \(G\)  is equivalent to
\(|s_1|+|s_2|+|s_3|^{1/2}+\log (1+|t_1|+|t_2|+|t_3|).\) 
